Lattices cubic

In a cubic crystal, the material properties are anisotropic, but there are a number of rotational symmetries. For example, rotations by multiples of 90° 

Unspecified components vanish. Thus, the three independent constants An, Ai2, and A44 remain. If the coordinate system is not parallel to the edges of the unit cell, a coordinate transformation of the elasticity tensor has to be used to find the components. In this case, the elasticity matrix takes a shape different from that in equation (2.34). 

Because the material properties are direction-dependent in a cubic crystal, they have to be stated together with the corresponding direction. According to the definition, the load direction has to be stated for Young s modulus Ei. Because the shear stress Tij and shear strain -y j have two indices, two indices are needed for the shear modulus Gij. Poisson s ratio relates strains in two directions. Here the second index j denotes the direction of the strain that causes the transversal contraction in the direction marked by the first index i eu = If the coordinate system is aligned with the axes 

As before, underscoring the indices indicates that no summation over this repeated index is done, see appendix A.4. 

It is sufficient to know the elastic constants in one coordinate system (for example, S , S 12, and 44) to calculate the properties in any other coordinate system. 

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Fig. XV-2. Adsorption loops in a cubic lattice. The diagrams depict the surface sites as seen from below. (From Ref. 4.)...

In tire simple version of tire lattice representation of proteins tire polypeptide chain is modelled as a sequence of connected beads. The beads are confined to tire sites of a suitable lattice. Most of tire studies have used tire cubic lattice. To satisfy tire excluded volume condition only one bead is allowed to occupy a lattice site. If all tire beads are identical we have a homopolymer model the characteristics of which on lattices have been extensively studied. 

Figure C2.5.2. Scaling of the number of MBS C(MES) (squares) is shown for the hydrophobic parameter = -0.1 and A = 0.6. Data were obtained for the cubic lattice. The pairs of squares for each represent the quenched averages for different samples of 30 sequences. The number of compact stmctures C(CS) and self-avoiding confonnations C(SAW) are also displayed to underscore the dramatic difference of scaling behaviour of C(MES) and C(CS) (or C(SAW)). It is clear that C(MES) remains practically flat, i.e. it grows no faster than In N.

In general, for a box which is positioned at a cubic lattice point n (= rij.L, riyL, n L) with n, Tiy, being integers) ... 

A cubic lattice is superimposed onto the solute(s) and the surrounding solvent. Values of the electrostatic potential, charge density, dielectric constant and ionic strength are assigned to each grid point. The atomic charges do not usually coincide with a grid point and so the... 

The summation is over the different types of ion in the unit cell. The summation ca written as an analytical expression, depending upon the lattice structure (the orij Mott-Littleton paper considered the alkali halides, which form simple cubic lattices) evaluated in a manner similar to the Ewald summation this typically involves a summc over the complete lattice from which the explicit sum for the inner region is subtractec... 

In Figure 8.19 is shown the X-ray photoelectron spectrum of Cu, Pd and a 60 per cent Cu and 40 per cent Pd alloy (having a face-centred cubic lattice). In the Cu spectrum one of the peaks due to the removal of a 2p core electron, the one resulting from the creation of a /2 core state, is shown (the one resulting from the 1/2 state is outside the range of the figure). 

Figure 8.19 X-ray photoelectron spectrum, showing core and valence electron ionization energies, of Cu, Pd, and a 60% Cu and 40% Pd alloy (face-centred cubic lattice). The binding energy is the ionization energy relative to the Fermi energy, isp, of Cu. (Reproduced, with permission, from Siegbahn, K., J. Electron Spectrosc., 5, 3, 1974)...

The direction of the alignment of magnetic moments within a magnetic domain is related to the axes of the crystal lattice by crystalline electric fields and spin-orbit interaction of transition-metal t5 -ions (24). The dependency is given by the magnetocrystalline anisotropy energy expression for a cubic lattice (33) ... 

A similar effect occurs in highly chiral nematic Hquid crystals. In a narrow temperature range (seldom wider than 1°C) between the chiral nematic phase and the isotropic Hquid phase, up to three phases are stable in which a cubic lattice of defects (where the director is not defined) exist in a compHcated, orientationaHy ordered twisted stmcture (11). Again, the introduction of these defects allows the bulk of the Hquid crystal to adopt a chiral stmcture which is energetically more favorable than both the chiral nematic and isotropic phases. The distance between defects is hundreds of nanometers, so these phases reflect light just as crystals reflect x-rays. They are called the blue phases because the first phases of this type observed reflected light in the blue part of the spectmm. The arrangement of defects possesses body-centered cubic symmetry for one blue phase, simple cubic symmetry for another blue phase, and seems to be amorphous for a third blue phase. 

With the Monte Carlo method, the sample is taken to be a cubic lattice consisting of 70 x 70 x 70 sites with intersite distance of 0.6 nm. By applying a periodic boundary condition, an effective sample size up to 8000 sites (equivalent to 4.8-p.m long) can be generated in the field direction (37,39). Carrier transport is simulated by a random walk in the test system under the action of a bias field. The simulation results successfully explain many of the experimental findings, notably the field and temperature dependence of hole mobilities (37,39). 

Material Cubic lattice constant, pm Band gap, eV Inttinsic carrier concentration, cm Relative dielectric constant, S Mobihty, Electrons cm"/(Vs) Holes... 

Table 1 Hsts the properties of several semiconductors relevant to device design and epitaxy. The properties are appropriate to the 2incblende crystal stmcture in those cases where hexagonal polytypes exist, ie, ZnS and ZnSe. This first group of crystal parameters appHes to the growth of epitaxial heterostmctures the cubic lattice constant, a the elastic constants, congment sublimation temperature, T. Eor growth of defect-free...

In the 2incblende stmcture the bond length is related to the cubic lattice parameter as (3a/4). ... 

Properties. Thallium is grayish white, heavy, and soft. When freshly cut, it has a metallic luster that quickly dulls to a bluish gray tinge like that of lead. A heavy oxide cmst forms on the metal surface when in contact with air for several days. The metal has a close-packed hexagonal lattice below 230°C, at which point it is transformed to a body-centered cubic lattice. At high pressures, thallium transforms to a face-centered cubic form. The triple point between the three phases is at 110°C and 3000 MPa (30 kbar). The physical properties of thallium are summarized in Table 1. 

Extensive computer simulations have been caiTied out on the near-surface and surface behaviour of materials having a simple cubic lattice structure. The interaction potential between pairs of atoms which has frequently been used for inert gas solids, such as solid argon, takes die Lennard-Jones form where d is the inter-nuclear distance, is the potential interaction energy at the minimum conesponding to the point of... 

How many atoms must be included in a three-dimensional molecular dynamics (MD) calculation for a simple cubic lattice (lattice spacing a = 3 x 10 ° m) such that ten edge dislocations emerge from one face of the cubic sample Assume a dislocation density of N = 10 m . ... 

Figure 2 A low energy conformation of a 27 mer lattice model on a 3 X 3 X 3 cubic lattice. (Adapted from Ref. 11.)...

To present briefly the different possible scenarios for the growth of multilayer films on a homogeneous surface, it is very convenient to use a simple lattice gas model language [168]. Assuming that the surface is a two-dimensional square lattice of sites and that also the entire space above the surface is divided into small elements, forming a cubic lattice such that each of the cells can be occupied by one adsorbate particle at the most, the Hamiltonian of the system can be written as [168,169]... 

In the case of the bond fluctuation model [36,37], the polymer is confined to a simple cubic lattice. Each monomer occupies a unit cube of the system and the bond length between the monomers can fluctuate. On the other... 

The bond fluctuation model (BFM) [51] has proved to be a very efficient computational method for Monte Carlo simulations of linear polymers during the last decade. This is a coarse-grained model of polymer chains, in which an effective monomer consists of an elementary cube whose eight sites on a hypothetical cubic lattice are blocked for further occupation (see... 

FIQ. 1 Sketch of the BFM of polymer chains on the three-dimensional simple cubic lattice. Each repeat unit or effective monomer occupies eight lattice points. Elementary motions consist of random moves of the repeat unit by one lattice spacing in one lattice direction. These moves are accepted only if they satisfy the constraints that no lattice site is occupied more than once (excluded volume interaction) and that the bonds belong to a prescribed set of bonds. This set is chosen such that the model cannot lead to any moves where bonds should intersect, and thus it automatically satisfies entanglement constraints [51],... 

Let us consider a simple self-avoiding walk (SAW) on a lattice. The net interaction of solvent-solvent, chain-solvent and chain-chain is summarized in the excluded volume between the monomers. The empty lattice sites then represent the solvent. In order to fulfill the excluded volume requirement each lattice site can be occupied only once. Since this is the only requirement, each available conformation of an A-step walk has the same probability. If we fix the first step, then each new step is taken with probability q— 1), where q is the coordination number of the lattice ( = 4 for a square lattice, = 6 for a simple cubic lattice, etc.). 

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